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The Riemann Hypothesis

C. Calderon

Departamento de Matematicas. Facultad de Ciencia y Tecnologa

Universidad del Pas Vasco. 48080 Bilbao, Spain

mtpcagac@lg.ehu.es

Monografas de la Real Academia de Ciencias de Zaragoza. 26: 130, (2004).

Abstract

Riemann proved (see [53], [37] and preferably Edwards book [15]), that (s)

has an analytic continuation to the whole plane apart from a simple pole at s = 1.

Moreover, he showed that (s) satisfies the functional equation

s/2(s/2)(s) = (1s)/2((1 s)/2)(1 s).

Riemann uses the functional equation to deduce an approximate formula for (x) =

#{p x;p prime}: that is

(x) Li(x) +Nn=2

(n)

n Li(x

1

n ), N > log x/ log2.

In the paper (see [53]), he stated also that (s) had infinitely many non trivial roots

and that it seemed probable that they all have real part 1/2.

1 Distribution of prime numbers

Among the positive integers there is a sub-class of special importance, that is, the class of

primes. Some questions about primes are: a) How many prime numbers are there? andb) How are the prime numbers distributed?

The answer to the question of how many prime numbers are there is given by the

theorem of Euclid (Elements, Book 9, Prop. 20): There exist infinitely many prime

numbers. In the proof of Euclid, to prove the theorem, it will suffice to prove that if

{p1, p2, , pn} is any finite set of primes, then we can find a prime number that is not inthe set.

As usual Z will denote the ring of integers, Q the field of rational numbers, R the field

of real numbers and C the field of complex numbers. Also we shall denote by N the set

of positive (natural) integers, not including 0.

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In 1737, Euler proved the existence of an infinity of primes by a new method, which

shows moreover that the series

pnp1n is divergent. Eulers work is based on the idea of

using an identity in which the primes appear on one side but not on the other. Stated

formally his identity is

p

{1 ps}1 =n=1

ns, (s > 1) (1)

where p ranges over all prime numbers. This formula results from expanding each of the

factors on the left

(1 x)1 = 1 + x + x2 + . . . , x = ps.As their product is a sum of terms of the form (p11 prr )s, where p1 = p2 = = pr,the formula (1) is deduced by using the fundamental theorem of arithmetic: Each natural

number n > 1 can be decomposed uniquely, up to order of the factors, as a product of

prime numbers.

From an old theorem of Nicole Oresme (1360), we have that: The harmonic seriesn=1 1/n = (1) is divergent.

Function (x). We introduce a function (x) which has become universally accepted

and means the number of primes not exceeding x. If p1 = 2, p2 = 3, , and pn denotethe nth prime number, then for each integer n we have (pn) = n. It follows from Euclid

s

theorem that (x) as x .Legendre. Experimentally, Legendre conjectured in 1798 and again in 1808 the hypoth-

esis that

(x) xlog x A(x) , limxA(x) = 1, 08366 . . .

Gauss. Gauss [20], in a letter to the astronomer Enke in 1849, stated that he had found

in his early years (at age 17, in 1792), that the number (x) of primes up to x and the

integralx

2dtlog t

are asymptotically equal. However, Gauss does not mention Eulers formula and he gives

no analytic basis for the approximation, which he presents only on the basis of extensive

computations. So, Gauss conjectured that

(x) xlog x

, x

This conjecture, now a theorem, is called the Prime Number Theorem.

Chebyshev. The first serious work on the function (x) is due to the Russian mathemati-

cian Chebyshev. On May 24, 1848, Chebyshev read at the Academy of St. Petersburg his

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first memoir on the distribution on prime numbers, later published in 1850. He proved,

using elementary methods, that for every > 0 there exists x0 > 0 such that if x > x0,

then

(C ) xlog x

< (x) < (C+ )x

log x

where C = 21/231/351/5301/30 < 1, C = (6/5)C. Moreover, Chebyshev showed that if

the limit

limx

(x)

x/ log x

exists, it must be equal to 1. He deduced also, that Legendres approximation of (x)

cannot be true, unless 1, 08366 is replaced by 1.

Dirichlet. In 1837, Dirichlet proved his famous theorem of the existence of infinitely

many primes in any arithmetic progression n

h (mod k), with h and k positive co-

primes integers. The main novelty in his proof consisted in making use of characters mod-

ulo k, that is, homomorphisms from the group (Z/kZ) of invertible residues (mod k)

into the multiplicative group of complex numbers of modulus 1, (see Tenenbaum [59] cap

II.8, for all these facts).

Riemann. Riemann considers the zeta-function defined by the generalized harmonic

series (s) =

ns, where the letter s denotes a complex variable s = + it > 1

its real part and t its imaginary part. If > 1 we can represent (s) by an absolutely

convergent infinite product, namely

(s) =p

{1 ps}1. (2)

The Riemann hypothesis appears in a paper over the number of primes less than a

given magnitude, published in 1859 by Riemann (see-preferably- Edwards book [15], also

[53], or [37]). It is the only work that Riemann published in number theory, but most of

Riemanns ideas have been incorporated in later to the work of many mathematicians.

In 1859 Dirichlet died and Riemann was appointed to the chair of mathematics atGottingen. A few days later Riemann was elected to the Berlin Academy of Sciences. He

had been proposed by Kummer, Borchardt and Weierstrass. Riemann, newly elected to

the Berlin Academy of Sciences had to report on their most recent research and he sent

a report on Uber die Anzahl der Prinzahlen unter einer gegebener Grose. Riemann style

is extremely difficult, his paper is extremely condensed and in particular [53], is probably

a summary of very extensive researches which he never found the time to expound at

grater length. The main purpose of the paper was to give estimates for the number of

primes less than a given number. In this paper he stated that (s) had infinitely manynon trivial roots and that it seemed probable that they all have real part 1/2. Many of

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the results which Riemann obtained in this paper were later proved by Hadamard, de la

Vallee-Poussin, Hardy and von Mangoldt.

2 Analytic continuation

The development of complex analysis was a central preoccupation of Riemanns and so it

comes as no surprise that from the beginning Riemann considered the zeta-function as an

analytic function. He first showed that (s) had an analytic continuation to the complex

plane as a meromorphic function which has only one singularity, a simple pole of residue

1 at s = 1.

Riemann derives his functional equation for (s) from the representation of the gamma-

function as an integral, for which Riemann still used the symbol

(s) = (s 1) = 0

euus1du. (3)

The gamma-function defined by (3) has meromorphic continuation to all of C, and is

analytic except at s = 0, 1, 2, , n, , where it has simple poles.If n is a positive integer, we have, on writing nx for u

(s)

ns=

0

enxxs1dx.

Hence

(s)(s) =

n=1

0

enxxs1dx =

0

xs1

dxex 1 , Re(s) > 1

if the inversion of the order of summation and integration can be justified. Next he

considers the integral

I(s) =

C

(x)s1ex 1 dx

where the contour C starts at infinity on the positive real axis, encircles the origin once in

the positive direction (counterclockwise), excluding the points 2in,n N and returnsup the positive real axis to +

. In the many-valued function (

x)s1 = e(s1) log(x) the

log(x) is determined in such a way that it is real for negative values of x.Thus Riemann deduced that for each s = 1

(s) = 2ss1 sin(s/2)(1 s)(1 s) (4)

(see the proof in [59] pag. 139 or [60], Chap. II, 2.3). The equation (4) gives a relationbetween (s) and (1 s) which, by making use of properties of the gamma-function,can be formulated as the statement that (s/2)s/2(s) remains unchanged when s is

replaced by 1

s. Indeed, on writing n2x for u in (s/2) we have

(s/2)s/2(s) =

0

(x)xs/21dx, ( > 1)

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where (x) =

n=1 en2x, and using the equation

2(x) + 1 = x1/2 [2(1/x) + 1]

for the theta-function of Jacobi, one obtains the relation

(s/2)s/2(s) =1

s(s 1) +

1

(x)(x(s/2)1 + x(1+s)/2)dx.

The last integral is convergent for all values of s, and so the formula holds, by analytic

continuation, for all values of s = 1. Now the right-hand side is unchanged if s is replacedby 1 s. Hence we can write the functional equation (4) in the more symmetric form

s/2(s/2)(s) = (1s)/2((1 s)/2)(1 s). (5)

(see [59], [11] or [15]). The symmetry of the functional equation and the poles at s = 0,and s = 1 ofs/2(s/2)(s), suggests the introduction of the function

(s) =1

2s(s 1)s/2(s/2)(s) (6)

that is an entire function that non vanishing in Re(s) > 1. Then from (5) and (6) we

have that the function (s) verifies the functional equation

(s) = (1 s). (7)

it also has no zeros in Re(s) < 0, apart from the trivial zeros at s = 2, 4, 6, . . .. Thusall the zeros have their real parts between 0 and 1.

Riemann considered (s) with s = 1/2 + it and stated that there is a product repre-

sentation

(t) = (0)

1 t

2

2

(8)

with zeros of(t) that correspond to the critical zeros of the zeta-function. Hadamard

studied ([21]) entire functions and their representations as infinite products. One conse-

quence is that the product formula (8) is valid.

Theorem. (Riemann) The zeta function (s) is analytic in the whole complex plane

except for a simple pole at s = 1 with residue 1. It satisfies the functional equation (4).

Riemann hypothesis. The non trivial zeros of the function (s) have real part equal to

1/2. In 1900 Hilbert included the resolution of Riemann hypothesis as the 8th problem

of his 23 problems for mathematicians of the twentieth century to work on. In the later

years of the nineteenth century several mathematicians had developed Riemanns work

to the point that Hadamard and C. de la Vallee Poussin independently of one another in

1896 proved the Prime Number Theorem.

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3 Riemanns explicit formula

Riemann gave an explicit formula for (x) in terms of the complex zeros of (s). In his

derivation of the formula, he uses complex integrals and the Cauchy residue theorem. The

starting point is the product representation (2) of (s) in Re(s) > 1. Thus, the relation

(2) implies that log (s) = log(1 ps). Expanding log(1 ps) and summing weobtain

log (s) =p

n

(1/n)pns, Re(s) > 1. (9)

With the assistance of these methods (as Riemann says), the number of prime numbers

that are smaller than x can now be determined. Let F(x) be equal to this number when

x is not exactly equal to a prime number; but let it be greater by 1/2 when x is a prime

number, so that, for any x at which there is a jump in the value in f(x)

F(x) =F(x + 0) + F(x 0)

2

and states: If in the identity (9) one now replaces pns by spn

xs1dx, (Re(s) > 1) for

each n N, then one obtainslog (s)

s=

0

f(x)xs1dx, Re(s) > 1 (10)

if one denotes

F(x) + (1/2)F(x1/2) + (1/3)F(x1/3) + . . .

by f(x). If we think of(s) as given and f(x) as required, then (10) represents an integral

equation to be solved for f(x).

He applies Fourier inversion to (10) to obtain that

f(x) =1

2i

a+iai

log (s)xsds

s, (a > 1). (11)

First, Riemann integrates by parts to obtain

f(x) = 12i

1log x

a+iai

d

ds

log (s)

s

xsds, (a > 1) (12)

From the relations (6) and (8), the following formula for log (s) is obtained

log (s) = log (0) +

log

1 + (s1/2)2

2

log(s/2 + 1) + (s/2) log log(s 1).

(13)

and substituting (13) in (12) one obtains his result

f(x) = Li(x)

Im>0

[Li(x) + Li(x1)] +

x

dt

t(t2 1)log t + log (0) (14)

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which is the Riemanns formula except that, as noted by Edwards in [15], log (0) equals

log 2, and

Li(x) = lim0

10

dt

log t+

x1+

dt

log t

=

x2

dt

log t+ 1, 04

To deduce the Riemans formula for (x), that is, for the number of primes less than

any given magnitude x, we observe that the functions f(x) and (x) are related by the

formula

f(x) = (x) +1

2(x

12 ) +

1

3(x

13 ) + + 1

n(x

1n ) + (15)

But if x1/n < 2 for any given x and n sufficiently large, then (x1n ) = 0. Thus the

series in (15) is finite. Riemann inverts this relation (15) by means of the Mobius inversion

formula (see

10.9 [15]) to obtain

(x) =n=1

(n)

nf(x

1n ). (16)

From (16) and (14) one obtains the Riemanns formula for (x)

(x) Li(x) +n=2

(n)

nLi(x

1n ). (17)

Riemann made in their paper the following assertions related to zeros of (s):

(1) There are infinitely many complex zeros of (s) which lie in the critical strip. Zeros

of the zeta-function in the critical strip are denoted = + i. Let T > 0, and let

N(T) denote the number of zeros of the function (s) in the region 0 Re(s) 1,0 < Im(s) T. That is

(T) = #{ = + i | 0 1, 0 < T}. (18)

According to Riemann, the number of zeros with imaginary parts between 0 and T is

aboutT

2log(

T

2) T

2.

(2) Riemann stated also, that the number of zeros of the zeta-function between this limits,

in the critical line Re(s) = 1/2, is about the same.

(3) If is a complex zero of (s), then the series | |2 converges and the series | |1 diverges.

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(4) The entire function (s) = s(s 1)(1/2)s(s/2)(s) can be written as Weierstrassproduct.

(5) All complex zeros of (s) lie on the critical line Re(s) = 1/2.

(6) The relation

f(x) = Li(x)

Li(x) +

x

dt

t(t2 1) log t log2, x > 1

where

Li(x) = Li(e logx), Li(ew) =

u+iv+iv

ez

zdz, w = u + iv,

Li(x) = limT+

||T

Lix

hods true.

A simpler variant of his formula is

(x) =nx

(n) = x

x

log2 1

2log(1 x2),

valid for x not a prime power (x = pm). Note that the Prime Number Theorem isequivalent to (x) x, x and that |x| = x; thus it was necessary to show that < 1 in order to conclude of Gausss conjecture, that is, the Prime Number Theorem.

Hypothesis (1), (3), (4) were proved by Hadamard [21]. The estimate of N(T) and

(6) were proved by Mangoldt [45].

4 Complex zeros of the zeta function

In order to prove the convergence of the product (8) Riemann needed to investigate the

distribution of roots of (s), then he begins to observing that if Re(s) > 1, then by the

Euler product, (s) = 0 (because a convergent infinite product can be zero only if one ofits factors is zero). Moreover if s = 1 + it we have the following result

Theorem ( Hadamard-de la Vallee Poussin). If t = 0, then (1 + it) = 0.As consequence, the zeta function has no zeros in the closed half plane Re(s) 1.If Re(s) < 0, then Re(1 s) > 1, the right-side in the functional equation (5) is not

zero, so the zeros must be exactly the poles of (s/2). Then, the zeros of (s) are

1. Simple zeros at the points s = 2, 4, 6, , which are called the trivial zeros

2. Zeros in the critical strip consisting of the complex zeros = + i with 0

Re() 1. In 1914, Hardy proved that there are infinitely many roots of(s) = 0on the line Re(s) = 1/2. (see [24] or [11]).

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Since (s) = (s), the zeros of(s) lie symmetrically with respect to the real axis, so,

it suffices to consider the zeros in the upper half of the critical strip. It is common to list

the complex zeros n = n + in, with n > 0, in order of increasing imaginary parts as

1 2 3 . Here zeros are repeated according to their multiplicity.

Van de Lune, te Riele and Winter [44] have determined that the first 1 .500.000.001 com-

plex zeros of (s) are all simple, lie on the critical line, and have imaginary part with

0 < < 545.439.823, 215.

(i) Let T > 0, and let N(T) denote the number of zeros of the function (s) in the region

0 Re(s) 1, 0 < Im(s) T. That is

N(T) = #{ = + i | 0 < 1, 0 < T}.

By the functional equation and the argument principle, one obtains

N(T) =T

2log(

T

2e) +

7

8+ S(T) + O(

1

T) (19)

where S(T) = 1

arg ( 12

+iT) log T with the argument obtained by continuous variationfrom 2 to 2 + iT, and thence to 1/2 + iT, along straight lines. It was conjectured by

Riemann and proved by von Mangoldt. This is the so called Riemann-von Mangoldt

formula, (the sign of I.M. Vinogradov is taken, as usual, in the sense O).If the Riemann hypothesis (henceforth RH for short) is true, then it is known that

S(T) =1

arg (

1

2+ iT) log T

log log T. (20)

(ii) Now, let N0(T) be the zero counting function

N0(T) = #{ = 1/2 + i | 0 < T},

that it, N0(T) counts the number of zeros of(s) in the critical line, up to height T. The

RH is equivalent to N(T) = N0(T) for all T.(iii) Let N(, T) the function which counts the number of zeros of(s) in the critical strip

up to height T, and to the right of-line.

A method to approach Riemanns hypothesis is to estimate for any given , 1/2 1, the number N(, T) of zeros = + i of(s), with and 0 < t T ( whereT is sufficiently large):

N(, T) = #{ = + i | , 0 < T}

1/2 1. In this case, the RH is equivalent to the property N(1/2, T) = 0 for all T.

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Estimates for N(, T) may be written in the form

N(, T) Ta()(1) logb T, b 0. (21)

In view of formula (19) one has a()(1 ) = 1 and b = 1 in (21) for 0 1/2, whilea()(1 ) 1 for > 1/2 and a()(1 ) is non increasing.

The hypothesis a() 2 in (21) is known as the density hypothesis. A. Selberg [57]has proved that

N(, T) T1 14 ( 12 ) log T (22)uniformly for 1/2 < 1. The RH is equivalent to N(, T) = 0 for > 1/2.Gaps between zeros. The most detailed study of the zeros on the critical line involves

estimates for the difference n+1

n between the ordinates of consecutive zeros s =

12

+i,

R. For a long time, the result of Hardy and Littlewood (1918) that n+1 n

1/4+n was the best. Later it was superseded, with the use of finer methods by Moser,

Balasubramanian, Karatzsuba, Ivic [31].

It is known that the average size of n+1 n is 2/ log n. Thus if

:= lim supn

(n+1 n) log n2

, := lim infn

(n+1 n)log n2

we have 1 , and it is known (unconditionally) that < 1 < . It is conjecturedthat = 0 and = , but assuming the generalized RH Conrey-Ghosh- Gonek [9]proved only that > 2.68. Also under the RH they had proved < 0.5172.

If (20) holds, then from the estimate (19) one obtains

N(T + H) N(T) > 0, for H = C/ log log T, C > 0, T T0.

Hence assuming the RH the following estimate holds

n+1

n

1

log log n.

Instead of working with the complex zeros of (s) on the critical line Re(s) = 1/2, it

is convenient to introduce the function

Z(t) = 1/2(1

2+ it)(

1

2+ it)

where (s) is given by (s) = 2ss1 sin(s/2)(1 s). As (s)(1 s) = 1 and (s) =(s), it follows that | Z(t) |=| ( 1

2+ it) |, Z(t) is even, and Z(t) = Z(t). Hence Z(t) is

real if t is real and the zeros of Z(t) correspond to the zeros of (s) on the critical line

Re(s) = 1/2.

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5 Consequences of the Riemann hypothesis

There are many equivalent forms for the Riemann hypothesis. The study and classification

of these assertions sharpens our understanding of the problem. Writing (t) = (1/2+it),

and as consequence of the functional equation (7), we obtain (t) = (

t) which is real

for real t an even function of t. Then RH is the assertion to all zeros of (t) are reals.

Riemann obtained that (t) verifies the relation

(t) = 4

1

d[x3/2(x)]

dxx1/4 cos

t

2log x

dx (23)

where (x) =

n=1 en2x. We can write (23) in the form

(t) = 2

0

(u)cos ut du

where (u) = 2

n=1(2n42e9t/2 3n2e5t/2)en2e2t . A necessary and sufficiently condi-

tion to of zeros of (t) are reals is

()()ei(+)xe()y( )2dd 0

for all x, y reals (see Chap X and XIV of [60]).

A. Speiser [58], proved that RH is equivalent to the non-vanishing of the derivate (s)

in the left-half of the critical strip 0 < < 1/2.

The order of zeta-function on the critical line. Hardy and Littlewood proved that

(1/2 + it) t 14+ for every positive , and t t0 > 0 (since (1/2 + it) = (1/2 it), tmay be assumed to be positive. H. Weyl improved the bound to t1/6+, Bombieri-Iwaniec

obtained 9/56, instead to 1/6, A. Watt improved the bound to 89/560 and M.N. Huxley

([29]) improved the bound to

(1/2 + it) t 89570+.

Lindelof conjectured that

(1/2 + it) t, t t0 > 0

for every positive . In 1912, Littlewood proved that the Lindelof hypothesis is true if the

RH is true. The converse theorem cannot be made.

For the function S(T) = 1

arg ( 12

+ iT) we have that the LH is equivalent to

T0

S(t)dt = o(log T), (T )

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while the RH implies much more

T0

S(t)dt log Tlog log T

, (T ).

The order of zeta-function on the 1-line. E.C. Titchmarsh, was proved ([60] theorems8.5 and 8.8), that each one of the inequalities

| (1 + it) |> A log log t, 1/ | (1 + it) |> A log log t

is satisfies for some arbitrary large values of t, if A is a suitable constant and considers

the question of how large the constant can be in the two cases.

On RH we have

e

lim sup

t

(1 + it)

log log t 2e

6

2e lim sup

t

1/(1 + it)

log log t 12

2e

where is Eulers constant, (see Titchmarsh [60] 8.9).

Riemann hypothesis and distribution of primes. We know that the zeta function

was introduced as an analytic tool for studying prime numbers and some of the most

important applications of the zeta functions belong to prime number theory. The equiv-

alence between the error term in the Prime Number Theorem and the real part of the

zeros of (s) is as follow

(x) =

x2

dt

log t+ O(x+)

or

(x) =pmx

logp = x + O(x log2 x)

is equivalent to (s) = 0 for Re(s) > , 1/2 < 1 (Th. 12.3 [31]). If we use thestrongest zero-free region then we deduce the results

(x) x

2dtlog t x1/2 exp(C(log x)3/5(log log x)1/5)

(x) x x1/2 exp(C(log x)3/5(log log x)1/5)but the Riemann hypothesis is equivalent to each of the following statements related to

prime numbers

(1) (x) =x

2dt

log t+ O(

x log x). [35]

(2) (x) =

pmxlogp = x + O(

x log2 x), x > 0.

In the problem related to the difference of two consecutive prime numbers, Pilts conjec-

tured that for every > 0, pn+1pn pn where pn denote the nth prime number, but this

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has never been proved. On the other hand, it is known that the relation pn+1pn logpnis certainly not true.

(3) The RH is equivalent to pn+1 pn p1/2n logpn.

The Mobius function. The Euler product formula for (s) implies that

1

(s)=p

{1 ps} =n=1

(n)

ns, s = + it, > 1.

The coefficient (n) is known as Mobius function.

It has the property

d|n (d) = 1, (n = 1) and 0 if n > 1, where d|n means that d isa divisor of n. Thus (n) verifies

(1) = 1, (n) = d|n,d 1).

Each one of the following properties is equivalent to the RH (see [60])

(4) M(x) =

nx (n) x1/2+, for all > 0,(5) the series

n=1 (n)/n

s is convergent, and its sum is 1/(s), for every s with

> 1/2.

The function w(n). Let (n) be the number of prime factors of the positive integer ncounted without multiplicity, and q N. D. Wolke proved that the RH is true if andonly if

nx

q(n) x

0j 12

log x

Rjq(log log x)(log x)j x1/2+

holds for every > 0, where Rjq(t) are certain polynomials such that deg R0q q, anddeg Rjq q 1, (j 1) (see [63]).Sum of divisors. Let (n) denote the sum of divisors of n, that is (n) = d|n d. G.Robin (see[54]) proved that RH is true, if and only if

(n) < en log log n

for large n, where is the Eulers constant. J.C. Lagarias [39], proved that under the RH

(n) < Hn + eHn log Hn, Hn =

nj=1

1/j, (n 2).

Bertrands postulate. In 1852, Chebyshev proved Bertrands postulate according towhich each interval (n, 2n], n 1, contains at least one prime. O. Romare, Y. Saoutier

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[55] under the RH proved that for all x 2 in the interval (x 85

x log x, x] there exist

a prime number.

Pythagorean triangles. The positive integers a,b,c are said to form a primitive

Pythagorean triples (a,b,c) if a

b, a,b,c are coprimes and a2 + b2 = c2. Let P(x)

denote the number of primitive Pythagorean triangles, whose area is less than x. In

1955, J. Lambek y L. Moser [Pacific J. Math. 5(1955), 73-83, MR 16, 796h] proved that

P(x) = c1x1/2 + O(x1/3). Muler, Nowak and Menzer ([50]), assuming the RH proved that

P(x) = c1x1/2 + c2x

1/3 + R(x), with R(x) x127/560+. The error term is connected withthe Mobius function and then with the zeros of the zeta function. W. Zhai [70] under the

RH deduced

R(x) x127/616(log x)963/308, 127/616 = 0.2061 . . .

A Divisor problem. Let a, b be positive coprime integers such that 1 a < b, and leta,b(x) be the error term for the asymptotic formula of

D(a, b; x) =

manbx, (m,n)=1

1, (1 a < b, (a, b) = 1.

Lu y Zhai [43], proved under the RH that

a,b(x) xa+6b

(2a+7b)(a+b)+, b 3a2

; a,b(x) xa+2b

(2a+2b)(a+b)+, b >3a

2.

The primitive circle problem. Let

P(x) =

m2+n2x,(m,n)=1

1

and let (x) = P(x) (6/)x. The primitive circle problem is to obtain an upperbound for (x). The best known unconditional result is

(x)

x1/2 exp(

c(log x)3/5(log log x)1/5).

Jie Wu ([68]), under the Riemann hypothesis proved that (x) x221/608+.Powerful numbers. Let k 2 be a fixed integer and let Nk denote the set of all positiveintegers n with the property that if a prime p | n, then pk | n. Thus the set of powerfulnumbers Nk contains 1 and the numbers whose canonical representation is n = p

11 prr ,

i k for all i = 1, 2, , r. We put fk(n) = 1 ifn Nk and fk(n) = 0 if n Nk andthe Dirichlet series representation is

Fk(s) =

n=1

fk(n)ns =p

(1 + pks

1 ps ), Re(s) > 1/k,

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then fk(n) is the characteristic function ofNk, (see E. Kratzel [36] Chap. 7). The problem

is to obtain an estimate for the number Nk(x) of k-full integers not exceeding x:

Nk(x) =nx

fk(n), k 2

the main term is deduced from the residues of Fk(s) at the simples poles s = 1/u,

u = k, k + 1, , 2k 1 and we have the following relation

Nk(x) =2k1t=k

ct,kx1/k + k(x), ct,k = Res

s=1/tFk(s)/s.

Let k = inf{k | k(x) xk}. A. Ivic showed that we would have under theassumption of the truth of Lindelof hypothesis, k

1/2k, but in special cases, it is

possible to prove much more. If k = 2, we have

F2(s) =(2s)(3s)

(6s).

Let N2(x) be denote the number of square-full integers n x, and 2(x) the errorterm. The best unconditional O-estimate is

2(x) = N2(x) (3/2)(3)

x12 (2/3)

(2)x

13 x 16 exp(c(log x) 35 (log log x) 15 ),

essentially due to P. T. Bateman and E. Grosswald [3] see Theorem 14.4 and page 438 in

A. Ivics book [31].

Under the RH, J. Wu [67] proved that 2(x) x(12/85)+, for any > 0, which can becompared with the complementary estimate (x) = (x1/10), due to R. Balasubramanian,

K. Ramachandra and M. V. Subbarao (Acta Arith. 50 (1988), no. 2, 107-118). Bateman

and Grosswald also noted that the statement 2(x) x, for any constant exponent < 1/6, is equivalent to some quasi-Riemann hypothesis, i.e. to the assertion that

(s)= 0 for all s with > 0 and 0 < 1.

A similar situation arises in case of cube-full integers.

Farey series and the RH. A sequence (xn) of real numbers is uniformly distributed

(mod 1) if and only if for every Riemann-integrable function f on [0, 1] one has

limN

1

N

nN

f({xn}) =1

0

f(x)dx

where

{x

}is the fractional part of x, ([10], Th. 3 Cahp. VIII). Let Fx = F[x] denote the

sequence of all irreducible fractions with denominators x, arranged in increasing orderof magnitude, that is

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Fx = {rk = ak/bk; 0 < ak bk x, (ak, bk) = 1, k = 1, 2, , (x)}, for any x 1, where(x) =

nx (n), ((n) = #{k n; (k, n) = 1} is the Euler function. F[x] is called

the Farey series of order x, (Note that the Farey series is not a series at all but a finite

sequence).

Since the Farey fractions are uniformly distributed (mod 1) we have that

limN

1

(N)

qN

qa=1

(a,q)=1

f(a/q) =

10

f(x)dx (24)

for every Riemann-integrable function f on [0, 1]. Relation (24) suggests the problem of

estimation of the error term

Ef(N) =qN

qa=1

(a,q)=1

f(a/q) (N) 1

0 f(x)dx.

Franel discovered a connection between Farey series and the Riemann hypothesis. He

proved ( [17]) that the estimation

k(x)

rk k

(x)

2 x2(1)+

for every > 0 is equivalent to (s)

= 0 for Re(s) > . Thus

RH is truth if and only if

k(x)

rk k

(x)

2 x1+.

Another version (Landau Vorlesungen ii, 167-177), is that the RH is equivalent to

the statement k(x)

| rk k(x)

| x 12+,

for all > 0 as x .M. Mikolas, (see [46]-[49]) proved that the RH is equivalent to the relation

(x)k=1

f(rk) (x)1

0

f(u)du x1/2+

for all > 0, where f(u) can be sin(u), cos(u), (|| = ) or a polynomial of degree 3.Huxley [28] has generalized Franels theorem to the case of Dirichlet L-functions and Fujii

[18] gives another equivalence with the RH in terms of the Farey series.

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Kanemitsu and Yoshimoto [33] showed that each one of the estimates are equivalent

to the RH

rk1/3

rk h(1/3)

2(x)

x1/2+,

rk1/4

rk h(1/4)

2(x)

x1/2+

where h(t) =

rkt1. Moreover, Kanemitsu and Yoshimoto obtain conditions equivalents

for a great class of functions (see also [34], [69]).

The Riesz sum. M. Riesz (see [60]) considers the function

f(x) =n=1

(1)n+1xn(n 1)!(2n) .

An application of the calculus of residues gives

f(x) = i2

a+i

ai

xsds(s)(2s) sin(s)

= i2

a+i

ai

(1 s)xsds(2s)

,

where 1/2 < a < 1. Taking a > 1/2 it follows that f(x) x1/2+. On the RH we canmove the line of integration to a = 1/4 + and obtain the estimation f(x) x1/4+.Conversely, by Mellins inversion formula

(1 s)(2s)

=

0

f(x)xs1dx

and if f(x)

x1/4+ holds, then the integral converges uniformly for

0

> 1/4, the

analytic function represented is regular for > 1/4 and the truth of the RH follows.

Hardy and Littlewood in 1918 [60] stated that RH holds if and only if

n=1

(x)nn!(2n + 1)

x1/4, x .

Nyman-Beurling criterion. The Riemann zeta function on may see as a Mellin trans-

form in the critical strip by means of the following formula

(s)s

=

0

(1/x)xs1dx, 0 < < 1

where (x) is the fractional part function (x) = x [x].B. Nyman and A. Beurling had the idea that it should be possible to translate the

Riemann hypothesis into a property of (x). Beurlings linear space N(0,1) consists of all

functions

x f(x) =n

k=1

ak(kx

), 0 < k 1

where the ak are constants such that,

nk=1 akk = 0. For 1 < p < , let Bp be the

closure of B in Lp(0, 1). In his thesis, Nyman [51] proved the following theorem:

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Theorem. The Riemann hypothesis is true if and only if N(0,1) is dense in L2(0, 1).

Beurling gives in his paper [4] a generalization of Nymans theorem

Theorem. Let1 < p < . The following properties are equivalent.(i) (s) has no zeros in the half plane Re(s) > 1/p

(ii) N(0,1) is dense in Lp(0, 1)

(iii) The characteristic function (0,1) is in the closure of N(0,1) in Lp(0, 1).

L. Baez-Duarte [1], stated a new version of Nyman-Beurling criterion:

Theorem. LetM(y) be denote the summatory function of (n), and let

Gn(x) =

n1

M()(1

x)

d

.

(1) If (x) = (0,1)(x)log x, then

Gn

p

0, as n

implies that (s)

= 0 para

Re(s) > 1/p. (2) If (s) = 0 for Re(s) > 1/p then Gn r 0, as n wheneverr (1, p).

Riemann -function and positivity criterion. The Riemann -function (6), is an

entire function of order one which is real-valued on the real axis and satisfies the relation

Re(s)(s)

> 0 when Re(s) > 1. The Riemann hypothesis is equivalent to the positivity

condition

Re

(s)

(s)

> 0, when Re(s) > 1/2.

(Hinkkanen [27]). J.C. Lagarias [38] defined an arbitrary discrete set in C which rep-

resents the set of zeros of an entire function f(z) counted with multiplicity. Lagarias

call admissible to the set if complex conjugate zeros and occur with the same

multiplicity, and the zeros satisfy the convergence condition

1+ | Re() |1+ | |2 < .

Theorem. ([38]) Let be an admissible zero of set inC. Then the following conditions

are equivalent

(1) All elements have Re() (2) The function f(s)/f(s) satisfies the positivity condition

Re(f(s)/f(s)) > 0, for Re(s) > .

If are the non trivial zeros of (s), and

hQ() = inf{Re

( + it)

( + it)

: < t < },

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then under the RH Lagarias proved that h() = ()/(), for > 1/2

This result is improved by R. Garunkstis [19], who obtained that if(s) = 0 for > a,1/2 a < 1 then h() = ()/() for > a.

Xian-Jin Li (see [40]) showed that a necessary and sufficient condition for the non

trivial zeros of the Riemann zeta function to lie on the critical line is that n = ((n 1)!)1(dn/dsn)(sn1 log (s)) |s=1 is non negative for every positive integer. Thus the RHholds if and only if n 0, for each n = 1, 2, 3, . . ..

He also showed that an identical result applies to the Riemann hypothesis for the

Dedekind zeta-function K(s) of a number field.

The number n can be written in terms of the complex zeros of (s) as

n =

(1 (1 1/)n

)

where the sum over is understood as

= limT

|Im()T .

Bombieri y Lagarias showed that Lis criterion follows as a consequence of a general

set of inequalities for an arbitrary multiset of complex numbers and therefore is not

specific to zeta function (see [5]).

Weil-Bombieri and RH. In 1952, A. Weil [62] gives a generalization of the Riemann

explicit formula and in the same paper Weil proved that from his formula one can definea quadratic functional whose positivity is equivalent to the Riemann hypothesis.

E. Bombieri [6], studies the Weil explicit formula. First of all, he provides a very

clear proof of the formula which relates the values of a smooth function f summed over

the primes to the sum of its Mellin transform f summed over the complex zeros of the

Riemann zeta-function.

Explicit Formula. Define f(x) = f(1/x)/x. If f(x) is any smooth complex-valued

function with compact support in (0, ) and Mellin transformf(x) =

0

f(x)xs1dx

then

f() =

0

f(x)dx +

0

f(x)dx n=1

(n)(f(n) + f(n))

(log 4 + )f(1)

1

f(x) + f(x) 2

xf(1)

x

x2 1dx.

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Bombieri next proves a strong version of Weils criterion for the Riemann hypothesis.

One takes a function f(x) = g g in the Weil formula, where is the multiplicativeconvolution of a function g and its transpose conjugate g.

Theorem (Bombieri). The Riemann Hypothesis holds if and only if

g()g(1 ) > 0

for every complex-valuedg(x) C0 ((0, )), not identically 0.

6 Generalizations of the Riemann hypothesis

Let {a(n)}n=1 be a sequence of complex numbers, and

n=1 a(n)ns, (s C) the associ-

ated Dirichlet series. When a(n)

1 we have the Riemann zeta function.

L-functions. Let k be a fixed positive integer. A Dirichlet character of conductor k is

a completely multiplicative function on the integers: that is (mn) = (m)(n) for every

pair of integers m, n, periodic with period k, (n + k) = (n), and such that (n) = 0

if (n, k) > 1. The principal character verifies (n) = 1 i f (n, k) = 1 and (n) = 0 i f

(n, k) > 1.

One can then define a L-function associated to (n) by

L(s, ) =

n=1

(n)ns

and as (n) is bounded, we have that the series is absolutely convergent for Re(s) > 1.

More than this is true, however: the series for L(s, ), ( non principal) converges for

Re(s) > 0. One has also an expansion as an Euler product

L(s, ) =p

(1 (p)ps)1.

It was introduced by Dirichlet for solving the problem of existence of infinity prime num-bers on arithmetic progressions.

Dirichlets theorem. There are infinitely many primes of the form kt + if and only if

and k are coprime integers (, k) = 1. (For instance, see Ellison Mendes-Frances book

for a proof [16]).

The Dirichlet series simplest after (s) is the Dirichlet L-function, L(s, 3), for the

non trivial character of conductor 3:

L(s, 3) =r=0

{(3r + 1)s (3r + 2)s}.

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This can be written as an Euler product

L(s, 3) =

p1 (mod 3)

{1 ps}1

p2 (mod 3)

{1 + ps}1.

The L-function L(s, 3) satisfies the functional equation

(s, 3) = (/3)(s+1)/2((s + 1)/2))L(s, 3) = (1 s, 3).

It is expected to have all of its nontrivial zeros on the critical line 1 /2.

One defines a character of conductor 4 as 4(n) = 0 if n is even; and

4(n) = (1)(n1)/2 if n is odd. Then

L(s, 4) =n=1

4(n)ns

= 1 13s

+ 15s

17s

+ , Re(s) > 1.

This can be written as an Euler product

L(s, 4) =

p1 (mod 4)

{1 ps}1

p3 (mod 4)

{1 + ps}1.

The L-function L(s, 4) also satisfies an analogue functional equation.

Theorem. Let be a non principal character, primitive of conductor k, (k > 1). The

L-function L(s, ) has an analytic continuation to the complex plane which is an entire

function of s. It verifies a functional equation

(s, ) = E()(1 s, )

where

(s, ) =

k

s+a2(

s + a

2)L(s, )

and a = 0 if (

1) = 1; a = 1 if (

1) =

1. The bar means complex conjugation and

E() = k1/2k1r=1

(r)e2ik .

For the proof, see for instance [11].

Since L(s, ) = 0 for > 1, (s, ) = 0 and (s, ) = 0 for > 1. By the functionalequation we obtain that (s, ) = 0 for < 0 so that all the zeros of (s, ) must lie inthe strip 0 1.

Corollary. If a = 0, the functionL(s, ) has simple zeros ats = 0, 2n, n N. If a = 1,then L(s, ) has simple zeros at s = (2n 1), n N. These are called the trivial zeros.

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As in the case of (s), one can deduce the existence of an infinity of non-real zeros

of L(s, ) either by using to the theory of entire functions, or by an estimate of the

Riemann-von Mangoldt type. One can prove that if N(T) denotes the number of zeros

{ = + i, 0 < < 1, 0 < T} of L(s, ) then

N(T) = T2

log T + AT + O(log T)

where A is a constant which depends on k, the modulus of character .

The theorem of the existence of an infinity of zeros of the critical line, proved by

Hardy for the (s), has been extended to a general class of Dirichlet series, including the

L-functions, by E. Hecke.

Generalized Riemann Hypothesis. The conjecture is that all the zeros of L-functions

are situated on the critical line.

Letdn

denote Kroneckers extension of the Legendre symbol for quadratic residues,

for n = 1, 2, . It is known that ifd is the discriminant of a quadratic number field (thatis, d is square-free and d 1 (mod 4); or d = 4N where N 2 or 3 (mod 4), andsquare-free), and d(n) =

dn

, then d is a real primitive Dirichlet character modulus

| d |. LetL(s, d) =

n=1

d(n)ns

be the associated Dirichlet L-function. It is an entire function (for d = 1) which satisfiesthe functional equation

(s, d) =

| d |s/2

s + ad

2

L(s, d) = (1 s, d)

where ad = 1 ifd < 0 and ad = 0 ifd > 0.

Zeta functions of number field. For a number field K of degree n, the zeta function

of K is

K(s) =

m=1

FK(m)ms

,

where FK(m) is the number of ideals whose norm is precisely m. This sum converges

for complex s with Re(s) > 1. Often K(s) is referred to as the Dedekind zeta function.

When K = Q, is just the Riemann zeta function: Q(s) = (s).

The function K(s) has an analytic continuation to the entire complex s-plane except

for a first order pole at s = 1. The residue of K(s) at s = 1 is

Ress=1K(s) = 2r1+r2

r2

hRw| D |

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wherer1 is the number of real conjugate fields ofK, 2r2 is the number of complex conjugate

fields of K, n is the degree of K, h is the class-number of K, R is the regulator of K, w

is the number of roots of unity in K, and D is the discriminant of K. There is also a

functional equation relating values at s = 1 to values at 1 s,

K(s) = K(1 s)

where

K(s) =

| D |22r2n

s/2r1(s/2)r2(s)K(s)

The function K(s) has no zeros in Re(s) > 1. It has trivial zeros with Re(s) < 0

of order r2 at 1, 3, 5, . . .; of order r1 + r2 at 2, 4, 6, . . .; and one zero of orderr1 + r2 1 at s = 0. All other zeros are in the critical strip. (See [65] chap. 6: GaloisTheory, Algebraic Number Theory, and Zeta Functions from H.M. Stark).

Extended Riemann Hypothesis. The Extended Riemann Hypothesis is the assertion

that the non trivial zeros of Dedekind zeta function of any algebraic number field lie on

the critical line.

Some consequences.

(1) Given the GRH, the Siegel-Walfisz theorem (see Th-5 [59], cap II.8) can be improvedto

(x; h, k) =nx

nh (mod k)

(n) =x

(k)+ O(

x log2 x), k x.

(2) Also we have

(x; h, k) =px

ph (mod k)

1 =x

(k)log x+ O(

x log x).

(3) D. Wolke (see [64] ), obtained the following equivalence for L-functions: Let 0, 1, 2, . . .

be the sequence of all Dirichlet characters (in which the principal character 0 occurs

only once), ordered which increasing moduli, and let {k}kN a sequence of real numbersk 0 for k which, in addition, satisfies the condition

kN

k log(k + 1) | log k |< .

Define the multiplicative function f : N C byn=1

f(n)ns = ((s))1k=1

(L(s, k))k , Re(s) > 1

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then Wolke proved that the GRH in true iff

nx

f(n) x1/2+, f or all > 0.

(4) The Goldbach conjecture. The three prime Goldbach conjecture states that every odd

integer 9 is a sum of three odd primes. The conjecture was proved under the GRH byDeshouillers-Effinger-te Riele-Zinoviev [14].

Quasi Riemann hypothesis. The term Quasi Riemann hypothesis related to a zeta-

function is used to mean that this zeta-function has no zeros in a half-plane > 0 for

some 0 < 1.

Modified Generalized Riemann Hypothesis. Say that an L-function satisfies the

Modified Great Riemann Hypothesis, if its zeros are all on the critical line or the realaxis.

The function (n) =

p|n 1 has normal order log log n and a famous theorem of

Turan bounds the variance of this distribution. Now let fa(n) be the smallest positive

integer m such that am 1 (mod n), with a relatively prime to n. Saidak ([56]) obtainedresults of Turan type for the function (fa(n)), thus assuming a quasi-Riemann hypothesis

for the Dedekind zeta functions of certain nonabelian number fields, he proves that for

each squarefree integer a

2,

px(p,a)=1

(fa(p)) log logp

2 (x)loglog x

nx(n,a)=1

(fa(n)) 1

2(log log n)2

2 x(log log x)3.

The Ramanujan zeta function. Let

L(s, ) =

n=1

(n)

ns , >

13

2 (25)

be the L-function attached to the Ramanujan function (n). The function (n) is multi-

plicative (proved by Mordell, 1917). The order of magnitude for (n) is |(n)| n11/2d(n),which was conjectured by Ramanujan in 1916 and proved by P. Deligne 60 years later

[13].

The Euler product corresponding to L(s, ) is

L(s, ) =p

(1 (p)ps

+ p112s

)1

, ( >

13

2 )

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and the functional equation, becomes

(2)s(s)L(s, ) = (2)s12(12 s)L(12 s, ).It has the trivial zeros s = 0, 1, 2, 3, but no other zeros for 11

2and 13

2.

The critical strip 0 < < 1 corresponding to the function (s) is the strip

11

2 < 1is the Hurwitz zeta function defined for Re(s) 1 by analytic continuation. For =arctan((

10 25 2)/(5 1)) it can be shown (see [12] or [60] cap. X), that f(s)

satisfies the functional equation

f(s) = 2 5s+ 12 (2)s1(1 s)cos( s2

)f(1 s).Moreover f(s) has an infinity of zeros on the line = 1/2 and it also has an infinity

of zeros in the half-plane Re(s) > 1, ( (s) = 0 in Re(s) > 1). Thus the function f(s)of Davenport and H. Heilbronn, satisfies a functional equation similar to the functionalequation for (s), but has no Euler product, and the analogue of the Riemann hypothesis

is FALSE for f(s).

7 Remarks.

The work of Riemann on the distribution of primes is thoroughly studied in Edwards

book [15], that I recommend strongly (a jewel), but a moderate knowledge of elementary

number theory and complex analysis is required of the reader. I suggest for example tostudy the Tenenbaums book (also the book of Ellison-Mendes). Another important text

is the great classic book of E.C. Titchmarh with an extensive treatment of the theory

of (s) up to 1951 and its second edition revised by D.R. Heath-Brown (see Chapters

XIII-XV for Riemann Hypothesis). A. Ivic [31] develops the theory of the Riemann zeta-

function and some applications, and the results proved in this text are unconditional, that

is, they do not depend on any hitherto unproved hypothesis.

There are so many results and conjectures having to do with the Riemann hypoth-

esis that it is very difficult to mention all of them. The reader interested on Riemannhypothesis can consult the Mathematical Reviews with about 1600 references (up today).

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References

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[2] Balazard, M. Completeness problems and the Riemann hypothesis; An annotated bibliogra-phy. Collection: Number theory for the Milleniunm (2002), 21-48.

[3] Bateman, P.T.; Grosswald, E. On a theorem of Erdos and Szekeres. Illinois-J.-Math. 2

(1958), 88-98.

[4] Beurling, A. A closure problem related to the Riemann zeta-function. Proc. Nat. Acad. Sci.

U.S.A. (1955), 41, 312-314.

[5] Bombieri, E. Lagarias, J.C. Complements to Lis criterion for the Riemann hypothesis. J.

Number Theory 77, 274-287, (1999).

[6] Bombieri, E. Remarks on Weils quadratic functional in the theory of prime numbers, I.

Att. Accad Naz. Licei (9) Mat. Appl. 11 (2000) n-3, 183-233.

[7] Calderon, C. La funcion zeta de Riemann. Rev. Real Academia de Ciencias de Zaragoza.

57, (2002), 67-87.

[8] Conrey,J.B. The Riemann Hypothesis. Not. Amer. Math. Soc. 50 (2003),no. 3, 341-353.

[9] Conrey,J.B.-Ghosh, A.-Gonek,S.M. Large gaps between zeros on the zeta function. Math-ematika, 33 (1986) 2, 212-338 (1987).

[10] Chandrasekharan, K. Introduction to analytic number theory.Springer Verlag New York

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[11] Chandrasekharan, K. Arithmetical functions. Springer Verlag New York 1969.

[12] Davenport, H.-Heilbrom, H. On the zeros of certain Dirichlet series, I, II.J. London Math.

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